Volumes in hyperbolic 5-space
An \(n\)-dimensional orthoscheme \(R\) is an \(n\)-simplex bounded by \(n + 1\) hyperplanes \(H_0, \ldots, H_n\) such that \(H_i\) and \(H_j\) are orthogonal if \(|i - j |> 1\). The \(n\) dihedral angles \(\alpha_i\) generated by the two hyperplanes \(H_{i - 1}\) and \(H_i\) are called the essential dihedral angles of \(R\). The two vertices of \(R\) opposite the hyperplanes \(H_0\) and \(H_n\) are called the principal vertices. In hyperbolic space, among all vertices of \(R\), at most the principal vertices may be points at infinity. If the two principal vertices of a hyperbolic orthoscheme are at infinity the orthoscheme is called doubly asymptotic. In 1954 (for the hyperbolic case) \textit{P. Müller} [`Über Simplexinhalte in nichteuklidischen Räumen'. Diss. Bonn (1954)] and in 1960 (for the hyperbolic and the elliptic case) the reviewer [Arch. Math. 11, 298-309 (1960; Zbl 0095.152)] had shown that, apart from lower order logarithms, the trilogarithm as a function of a single variable suffices to express volumes of orthoschemes (and therefore furthermore for polytopes) of dimension five. In a former paper [Comment. Math. Helv. 67, No. 4, 648-663 (1992; Zbl 0771.51011)] the author had considered the volume of special doubly asymptotic 5-orthoschemes, here given once more. In this paper the author presents an explicit volume formula for doubly asymptotic 5-orthoschemes. The formula is expressed in terms of polylogarithms of order at most three in the essential dihedral angles. By a result of \textit{C. H. Sah} [Math. Scand. 49, 181-210 (1981; Zbl 0496.52003)] and \textit{H. E. Debrunner} [Geom. Dedicata 33, No. 2, 123-152 (1990; Zbl 0699.51012)] every hyperbolic polytope of odd dimension can be equidissected into doubly asymptotic orthoschemes. Therefore the volume of a polytope in hyperbolic 5-space can be calculated by means of the volume of doubly asymptotic 5-orthoschemes. To prove her volume formula the author uses Schläfli's differential formula and Lobachevskij's result in the three dimensional case. In an appendix we find a summary on polylogarithms, in particular, for order two and three, and furthermore, how the characteristic volume integral can be represented in terms of trilogarithm functions.
scientific article; zbMATH DE number 800442
scientific article